Chapter 3: Plasticity

Common tests used to determine the monotonic strength of metals. (a) Uniaxial tensile test.(b) Upsetting test. (c) Three-point bending test. (d) Plane-strain tensile test. (e) Plane-straincompression (Ford) test. (f) Torsion test. (g) Biaxial test.

Tests for Mechanical Strength of Materials

A servohydraulicuniversal testing machine linked toa computer. (Courtesy of MTSSystems Corp.)

Mechanical Testing: Servohydraulic Machine

Stress–strain curves forAISI 1040 steel subjected todifferent heat treatments; curvesobtained from tensile tests.

Stress-Strain Curves at Different Heat Treatments

Idealized shapes ofuniaxial stress–strain curve. (a)Perfectly plastic. (b) Idealelastoplastic. (c) Ideal elastoplasticwith linear work-hardening. (d)Parabolic work-hardening (σ =σo + Kεn).

Uniaxial Stress-Strain Curve

Ludwik-Hollomon equation

Plasticity

Voce equation

Johnson-Cook equation

Schematicrepresentation of the change inPoisson’s ratio as the deformationregime changes from elastic toplastic.

True Stress and True Strain Curve

True- andengineering-stress–strain curvesfor AISI 4140 hot-rolled steel. R.A. is reduction in area.

Stress-Strain Curve

Engineering- (or nominal-) stress–strain curves (a) without and (b) with a yieldpoint.

Engineering Stress and Engineering Strain

Tensile specimen being tested; arrows show onset of necking.

Tensile tests

Log dσ/dε versus log εfor stainless steel AISI 302.(Adapted with permission from A.S. de S. e Silva and S. N. Monteiro,Metalurgia-ABM, 33 (1977) 417.)

Work hardening vs. Strain

Correction factor fornecking as a function of strain inneck, ln(A0/A), minus strain atnecking, εu. (Adapted withpermission from W. J. McGregorTegart, Elements of MechanicalMetallurgy (New York: MacMillan,1964), p. 22.)

Necking

Stress–strain curves for Fe–0.003% C alloy wire, deformed to increasingstrains by drawing; each curve is started at the strain corresponding to the priorwire-drawing reduction. (Courtesy of H. J. Rack)

(a) Effect of strain rateon the stress–strain curves forAISI 1040 steel. (b) Strain-ratechanges during tensile test. Fourstrain rates are shown: 10−1,10−2, 10−3, and 10−4 s−1.

Strain Rate Effects

(a) Compressionspecimen between parallel platens.(b) Length inhomogeneity inspecimen.

Plastic Deformation in Compressive Testing

(a) Stress–strain(engineering and true) curves for70–30 brass in compression. (b)Change of shape of specimen andbarreling.

Stress-Strain Curve for Compression

(a) Distortion of Finite Element Method (FEM) grid after 50% reduction inheight h of specimen under sticking-friction conditions. (Reprinted with permission from H. Kudo and S. Matsubara, Metal Forming Plasticity (Berlin: Springer, 1979),p. 395.) (b) Variation in pressure on surface of cylindrical specimen beingcompressed.

Finite Element Method

The Bauschinger effect.

Ratio of compressiveflow stress (0.2% plastic strain) andtensile flow stress at differentlevels of plastic strain for differentsteels. (After B. Scholtes, O.V¨ohringer, and E. Macherauch,Proc. ICMA6, Vol. 1 (New York:Pergamon, 1982), p. 255.)

Bauschunger Effect

Schematic of thedifferent types of stress–straincurves in a polymer.

Effect of strain rateand temperature on stress–strain curves.

Plastic Deformation of Polymers

Schematic of neckingand drawing in a semicrystalline polymer.

Glassy Polymers

(a) Neck propagationin a sheet of linear polyethylene.(b) Neck formation andpropagation in a specimen, shown in schematic fashion.

Neck Propagation in Polyethylene

Metallic Glasses

Compressionstress–strain curves forPd77.5CU6Si16.5. (Adapted withpermission from C. A. Pampillo and H. S. Chen, Mater. Sci. Eng., 13 (1974) 181.)

Plastic Deformation of Glasses

Shear stepsterminating inside material afterannealing at 250◦C/h, produced by (a) bending and decreased by (b)unbending. MetglasNi82.4Cr7Fe3Si4.5B3.1 strip. (Courtesy of X. Cao and J. C. M. Li.)

Shear Steps

(a) Gilman model ofdislocations in crystalline andglassy silica, represented bytwo-dimensional arrays of polyhedra. (Adapted from J. J. Gilman, J. Appl. Phys. 44 (1973)675) (b) Argon model of displacement fields of atoms (indicated by magnitude anddirection of lines) whenassemblage of atoms is subjected to shear strain of 5 × 10−2, inmolecular dynamics computation. (Adapted from D. Deng, A. S.Argon, and S. Yip, Phil. Trans. Roy. Soc. Lond. A329 (1989) 613.)

Dislocations

Viscosity ofsoda–lime–silica glass and ofmetallic glasses (Au–Si–Ge,Pd–Cu–Si, Pd–Si, C0P) as afunction of normalizedtemperature. (Adapted from J. F.Shakelford, Introduction to MaterialsScience for Engineers, 4th ed.(Englewood Cliffs, NJ: PrenticeHall, 1991), p. 331, and F. Spaepenand D. Turnbull in Metallic Glasses,ASM.) 1P=0.1 Pa · s.

Viscosity of threeglasses as a function oftemperature. 1 P=0.1 Pa · s.

Viscosity of Glass

Rankine, Tresca, and von Mises

Maximum-stress Criterion

Maximum-Shear-Stress Criterion

Maximum-Distortion-Energy Criterion

(a) Comparison of theRankine, von Mises, and Trescacriteria. (b) Comparison of failurecriteria with test. (Reprinted withpermission from E. P. Popov,Mechanics of Materials, 2nd ed.(Englewood Cliffs, NJ:Prentice-Hall, 1976), and G.Murphy, Advanced. Mechanics ofMaterials (New York: McGraw-Hill,1964), p. 83.)

Comparison of the Rankine, von Mises, and Tresca

Displacement of theyield locus as the flow stress of thematerial due to plasticdeformation. (a) Isotropichardening. (b) Kinematichardening.

Displacement of the Yield Locus

Tensile and Compressive Strength of Al2O3

Mohr-Coulomb failure criterion

Griffith Failure Criterion

McClintock-Walsh Crtierion

(a) Simple model for solid with cracks. (b) Elliptical flaw in elastic solid subjected to compression loading. (c) Biaxial fracture criterion for brittle materials initiated from flaws without (Griffith)and with (McClintock and Walsh) crack friction.

Failure Criteria for Brittle Material

Translation of vonMises ellipse for a polymer due tothe presence of hydrostatic stress.(a) No hydrostatic stress, (b) withhydrostatic stress.

von Mises Ellipse

Envelopes definingshear yielding and crazing for anamorphous polymer under biaxialstress. (After S. S. Sternstein and L.Ongchin, Am. Chem. Soc., Div. ofPolymer Chem., Polymer Preprints, 10(1969), 1117.)

Shear Yielding and Crazing for Amorphous Polymer

Failure envelope for unidirectional E-glass/epoxy composite under biaxialloading at different levels of shear stress. (After I. M. Daniel and O. Ishai, Engineering Mechancis of Composite Materials (New York: Oxford University Press, 1994), p. 121.)

Failure Envelope

Plane-stress yield locifor sheets with planar isotropy ortextures that are rotationallysymmetric about the thicknessdirection, x3. (Values of R indicatethe degree of anisotropy =σ2/σ1.)

Plane-Stress Yield Loci for Sheets with Planar Isotropy

Hardness for Steel

Comparison of the impression sizes produced by various hardness tests onmaterial of 750 HV. BHN = Brinell hardness number, HRC = Rockwell hardnessnumber on C scale, HRN = Rockwell hardness number on N scale, VPN = Vickershardness number. (Adapted with permission from E. R. Petty, in Techniques of MetalsResearch, Vol. 5, Pt. 2, R. F. Bunshah, ed. (New York: Wiley-Interscience, 1971), p. 174. )

Hardness Tests

Impression caused byspherical indenter on metal plate.

Impression

Procedure in usingRockwell hardness tester.(Reprinted with permission fromH. E. Davis, G. E. Troxel, and C. T.Wiscocil, The Testing and Inspectionof Engineering Materials, (NewYork: McGraw-Hill, 1941), p. 149.)

Rockwell Hardness Tester

Vickers Hardness Test

Relationships Between Yield Stress and Hardness

(a) Hardness–distanceprofiles near a grain boundary inzinc with 100-atom ppm of Al andzinc with 100-atom ppm of Au(1-gf load). (b) Soluteconcentration dependence ofpercent excess boundaryhardening in zinc containing Al, Au,or Cu (3-gf load). (Adapted withpermission from K. T. Aust, R. E.Hanemann, P. Niessen, and J. H.Westbrook, Acta Met., 16 (1968)291.)

Hardness Distance Profile

Some of the details ofthe Knoop indenter, together withits impression.

Knoop Indenter

A schematic of ananoindenter apparatus.

Nanoindenter apparatus

An impression madeby means of Berkovich indenter ina copper sample. (From Deng,Koopman, Chawla, and Chawla,Acta Mater., 52 (2004) 4291.) (a)An atomic force micrograph,which shows very nicely thetopographic features of theindentation on the sample surface.The scale is the same along thethree axes. (b) Berkovichindentation as seen in an SEM.

Topographic Feature of the Berkovich Indentation

A schematicrepresentation of load vs. indenterdisplacement.

Load vs. Indentation Displacement

Simple formabilitytests for sheets. (a) Simple bendingtest. (b) Free-bending test. (c)Olsen cup test. (d) Swift cup test.(e) Fukui conical cup test.

Simple Formability Tests for Sheets

“Ears” formed indeep-drawn cups due to in-planeanisotropy. (Courtesy of Alcoa,Inc.)

Plastic Anisotropy

Effect of “fibering” on formability. The bending operation is often an integralpart of sheet-metal forming, particularly in making flanges so that the part can beattached to another part. During bending, the fibers of the sheet on the outer side ofthe bend are under tension, and the inner-side ones are under compression. Impuritiesintroduced in the metal as it was made become elongated into “stringers” when themetal is rolled into sheet form. During bending, the stringers can cause the sheet to failby cracking if they are oriented perpendicular to the direction of bending (top). If theyare oriented in the direction of the bend (bottom), the ductility of the metal remainsnormal. (Adapted with permission from S. S. Hecker and A. K. Ghosh, Sci. Am., Nov.(1976), p. 100.)

Fibering

Sheet specimensubjected to punch–stretch testuntil necking; necking can be seenby the clear line. (Courtesy of S. S.Hecker)

Punch-Stretch Test

Schematic of sheetdeformed by punch stretching. (a)Representation of straindistribution: ε1, meridional strains;ε2, circumferential strains; h, cupheight. (b) Geomety of deformedsheet.

Punch-Stretch Test

Construction of aforming-limit curve (orKeeler–Goodwin diagram).(Courtesy of S. S. Hecker.)

Forming-Limit Curve

Different strainpatterns in stamped part. (Adaptedfrom W. Brazier, Closed Loop, 15,No. 1 (1986) 3.)

Different Strain Patterns in Stamped Part

Stress vs. Strain Rate for Slow-Twitch and Fast Twitch Muscles

Stress–strain responsefore a number of biologicalmaterials.

Strength of Biological Materials

Stress–strain responsefor elastin; it is the ligamentumnuchae of cattle (Adapted from Y.C. Fung and S. S. Sobin, J. Biomech.Eng., 1103 (1981) 121. Also in Y.C. Fung, Biomechanics: Mechanicaproperties of Living Tissues(NewYork: Springer, 1993) p. 244.)

Stress-Strain Response of Elastin

Tensile andcompressive stress–strain curvesfor cortical bone in longitudinaland transverse directions.(Adapted from G. L. Lucas, F. W.Cooke, and E. A. Friis, A Primer onBiomechanics (New York: Springer,1999).)

Stress-Strain Response of Cortical Bone

Strain-ratedependence of tensile response ofcortical bone. (Adapted from J. H.McElhaney, J. Appl. Physiology,21(1966) 1231.)

Strain Rate Response of Cortical Bone